All Packages  Class Hierarchy  This Package  Previous  Next  Index

Class math.topol.Braid.braid

java.lang.Object
   |
   +----math.topol.Braid.braid

public class braid

extends Object

Version:

$Id: braid.java,v 1.11 1999/02/12 20:36:12 djun Exp djun $
Copyright (c)1998-1999 Djun Kim. The author reserves all rights.

crosslist

The braid word: a list of elementary generators.

zerovect

braid(int)

Creates an empty braid with the given number of strands.

braid(int[])

Creates a braid from the given word in the standard generators (elementary braids), given as an int[] array.

braid(intArray)

Creates a braid from the given word in the standard generators (elementary braids), given as an intArray.

braid(String)

Creates a braid from the given String in the standard generators (elementary braids).

ArtinDecomposition(braid)

Returns a Vector of intArrays, which represents the fully combed form of the given (pure!) braid.

binomial(int, int)

Get small binomial coefficient ($n$ choose $k$) as an int, for small values of $k$.

comb(braid)

Returns a braid, which represent the fully combed form of the given (pure!) braid.

dbinomial(int, int)

Returns large binomial coefficient $(-1)^{k1} \times k! / k1! (k-k1)!$

embed()

Returns this braid naturally embedded in a braid with one more strand.

factorialExpansion(int, int)

Returns the factorial expansion of $g$ down to the $(r1 - 1)$-th position.

freeCanonicalForm(braid)

Returns a braid which is equivalent to this braid, but which is presented as a product of free group generators.

freeReduce()

Reduces this braid, in the free group.

getLength()

Returns the length of this braid.

getNumStrands()

Returns the braid index (number of strands) of this braid.

getPermutation()

Returns the permutation of this braid.

init()

Initializes the variables required for the polynomial computations.

inverse()

Returns the inverse of this braid.

isPure(braid)

Returns true if and only if the given braid is a pure braid.

random(int, int, Random)

Returns a pseudo-random braid of given index, of length less than or equal to the given length.

random(int, Random)

Returns a pseudo-random braid of given index.

retract(int)

Returns the k-fold retraction of this braid, that is, the braid resulting from deleting the last k strands.

seedRandom()

Initializes the pseudo-random braid generator with a "random" value (given by the system clock).

seedRandom(long)

Initializes the pseudo-random braid generator with the given seed value of type long.

setBraid(intArray)

Sets this braid to the given braidword, expressed as an intArray.

setNumStrands(int)

Sets the braid index (number of strands) of this braid.

thread()

Creates a representation of this braid as a vector of crossings, with each crossing recording its index, which strands are involved, and whether it is a positive or a negative crossing.

times(braid)

Returns the product (this braid)*(b).

toFreeGenList()

Returns a list (intArray) representation of this braid, as a sequence of free generators.

toFreeGenTeXForm()

Returns a String representation of this braid, written as a string of free generators.

toString()

Returns a string representation of this braid.

toTeXForm()

Returns a String representation of this braid, formatted as TeX input.

two_variable(PrintWriter, StringBuffer)

Computes the two-variable polynomial from a braid.

untwist(braid. disk)

Untwists an "innermost" twist of form $\sigma_k^{\mp 1} \cdots$ (other generators involving strand $n$) $\cdots \sigma_{\ell}^{\pm 1}$, where $\sigma_k$ and $\sigma_{\ell}$ involve strands $i$ and $j$, and are opposite in sign.

 public intArray crosslist

The braid word: a list of elementary generators.

 public static final int zerovect[]

 public braid(int numStrands)

Creates an empty braid with the given number of strands.

 public braid(int braidword[])

Creates a braid from the given word in the standard generators (elementary braids), given as an int[] array.

 public braid(intArray braidword)

Creates a braid from the given word in the standard generators (elementary braids), given as an intArray.

 public braid(String braidword)

Creates a braid from the given String in the standard generators (elementary braids).

 public int getLength()

Returns the length of this braid.

 public void setBraid(intArray braidword)

Sets this braid to the given braidword, expressed as an intArray.

 public static boolean isPure(braid b)

Returns true if and only if the given braid is a pure braid.

 public int getNumStrands()

Returns the braid index (number of strands) of this braid.

 public void setNumStrands(int numStrands)

Sets the braid index (number of strands) of this braid.

 public braid embed()

Returns this braid naturally embedded in a braid with one more strand.

 public braid retract(int k)

Returns the k-fold retraction of this braid, that is, the braid resulting from deleting the last k strands. Assume that this braid has been threaded. Returns null if such a retraction is undefined.

 public braid inverse()

Returns the inverse of this braid.

 public braid times(braid b)

Returns the product (this braid)*(b). Doesn't attempt to reduce the result.

 public void freeReduce()

Reduces this braid, in the free group. Also removes any zero entries.

 public void untwist(braid. disk D)

Untwists an "innermost" twist of form $\sigma_k^{\mp 1} \cdots$ (other generators involving strand $n$) $\cdots \sigma_{\ell}^{\pm 1}$, where $\sigma_k$ and $\sigma_{\ell}$ involve strands $i$ and $j$, and are opposite in sign.

Parameters:

startindex - points to the crossing $\sigma_k$

endindex - points to the crossing $\sigma_{\ell}$.

 public Vector ArtinDecomposition(braid pb)

Returns a Vector of intArrays, which represents the fully combed form of the given (pure!) braid.

 public braid comb(braid pb)

Returns a braid, which represent the fully combed form of the given (pure!) braid.

 public braid freeCanonicalForm(braid inbraid)

Returns a braid which is equivalent to this braid, but which is presented as a product of free group generators. This assumes that this braid is in fact an element of the normal subgroup of the pure n-strand braid group which is the kernel of the retraction map, in the form of a product of the free (Artin) generators. No checking is done on input!

See Also:

retract

 public int[] getPermutation()

Returns the permutation of this braid. The permutation is given as an array of numStrands integers, which represent the permutation of the braid on the array [0, ... , numStrands].

 public static Random seedRandom(long seed)

Initializes the pseudo-random braid generator with the given seed value of type long. This returns an instance of class Random, which is a "handle" to a random number generator.

 public static Random seedRandom()

Initializes the pseudo-random braid generator with a "random" value (given by the system clock).

 public static braid random(int index,
                            int length,
                            Random randgen)

Returns a pseudo-random braid of given index, of length less than or equal to the given length. The braid is generated by invokation of methods in java.util.Random. A random number generator must be supplied, for example by a call to seedRandom. Note: the value of index should satisfy 1 < index < 128.

 public static braid random(int index,
                            Random randgen)

Returns a pseudo-random braid of given index. The braid is generated by calls to the methods of java.util.Random. A random number generator must be supplied, for example by a call to seedRandom. Note: the value of index should satisfy 1 < index < 128. The length of the braid will be a random number between 0 and 128.

 public void thread()

Creates a representation of this braid as a vector of crossings, with each crossing recording its index, which strands are involved, and whether it is a positive or a negative crossing.

 public String toString()

Returns a string representation of this braid.

Overrides:

toString in class Object

 public String toTeXForm()

Returns a String representation of this braid, formatted as TeX input.

 public intArray toFreeGenList()

Returns a list (intArray) representation of this braid, as a sequence of free generators. This assumes that this braid has already been put into the appropriate form by a call to freeCanonicalForm. The generators are given by $$ x_i = \sigma_{k} \sigma_{k-1} \cdots \sigma_{i}^2 \sigma_{i+1}^{-1} \cdots \sigma_{k}^{-1} $$ which is represented as $i$. The inverse is given by $-i$.

 public String toFreeGenTeXForm()

Returns a String representation of this braid, written as a string of free generators. This assumes that this braid has already been put into the appropriate form by a call to freeCanonicalForm. The generators are given by $$ x_i = \sigma_{k} \sigma_{k-1} \cdots \sigma_{i}^2 \sigma_{i+1}^{-1} \cdots \sigma_{k}^{-1} $$

 public void init()

Initializes the variables required for the polynomial computations. This method must be called before any polynomial may be calculated. Since it initializes an structures which are factorially large in the size of the input, it may consume significant computational resources. Once the global tables are initiallized, however, subsequent polynomial computations are fast.

 public void two_variable(PrintWriter out,
                          StringBuffer braidstr)

Computes the two-variable polynomial from a braid.

 public static double dbinomial(int k,
                                int k1)

Returns large binomial coefficient $(-1)^{k1} \times k! / k1! (k-k1)!$

 public static int[] factorialExpansion(int g,
                                        int r1)

Returns the factorial expansion of $g$ down to the $(r1 - 1)$-th position.

 public static int binomial(int v,
                            int v1)

Get small binomial coefficient ($n$ choose $k$) as an int, for small values of $k$.

All Packages  Class Hierarchy  This Package  Previous  Next  Index